ACT Math Test Practice

The categories are listed below with the number of questions fromeach category: ■ Pre-Algebra 14 questions ■ Elementary Algebra 10 questions ■ Intermediate Algebra 9 questions ■ Coordina

 O v e r v i e w : A b o u t t h e A C T M a t h Te s t

The 60-minute, 60-question ACT Math Test contains questions from six categories of subjects taught in mosthigh schools up to the start of 12th grade The categories are listed below with the number of questions fromeach category:

■ Pre-Algebra (14 questions)

■ Elementary Algebra (10 questions)

■ Intermediate Algebra (9 questions)

■ Coordinate Geometry (9 questions)

■ Plane Geometry (14 questions)

taught in algebra class You will, however, need a strong foundation in all the subjects listed on the previouspage in order to do well on the math test You may use a calculator, but as you will be shown in the follow-ing lessons, many questions can be solved quickly and easily without a calculator.

Essentially, the ACT Math Test is designed to evaluate a student’s ability to reason through math lems Students need to be able to interpret data based on information given and on their existing knowledge

prob-of math The questions are meant to evaluate critical thinking ability by correctly interpreting the problem,analyzing the data, reasoning through possible conclusions, and determining the correct answer—the onesupported by the data presented in the question

Four scores are reported for the ACT Math Test: Pre-Algebra/Elementary Algebra, Intermediate bra/Coordinate Geometry, Plane Geometry/Trigonometry, and the total test score

1 If a student got 95% of the questions on a 60-question test correct, how many questions did the

stu-dent complete correctly?

3 What is the value of x in the equation −2x + 1 = 4(x + 3)?

6 Joey gave away half of his baseball card collection and sold one third of what remained What fraction

of his original collection does he still have?

15 AC and BC are both radii of circle C and have a length of 6 cm The measure of ∠ACB is 35° Find thearea of the shaded region.

18 The equation of line l is y = mx + b Which equation is line m?

19 If Mark can mow the lawn in 40 minutes and Audrey can mow the lawn in 50 minutes, which

equa-tion can be used to determine how long it would take the two of them to mow the lawn together?

 P r e t e s t A n s w e r s a n d E x p l a n a t i o n s

1 Choice a is correct Multiply 60 by the decimal equivalent of 95% (0.95) 60 × 0.95 = 57.

2 Choice f is correct Look at the pattern below.

6 Choice h is correct After Joey sold half of his collection, he still had half left He sold one third of the

half that he had left (13×12= 16), which is 16of the original collection In total, he gave away 12and sold

16, which is a total of23of the collection (12+ 16= 36+ 16= 46= 23) Since he has gotten rid of23of the lection,13remains

col-7 Choice a is correct Break up 40 into a pair of factors, one of which is a perfect square.

40 = 4 × 10

The problem asked for the measure of ∠RST which is 2x Since x is 23, 2x is 46°.

10 Choice j is correct Substitute the given values into the equation and solve for h.

13 Choice d is correct Find the equation of the line containing (2, −3) and (6, 1) First, find the slope.

Substitute the ordered pairs into the equations The pair that makes the equation true is on the line

When (7, 2) is substituted into y = x − 5, the equation is true.

5 = 7 − 2 is true

14 Choice f is correct Triangle MNP is a 3-4-5 right triangle The height of the triangle is 4 and the base

is 3 To find the area use the formula A = b2h

A = (3)2(4)= 122= 6

The area of the triangle is 6 square inches

15 Choice d is correct Find the total area of the circle using the formula A = πr2

A = π(6)2= 36π

A circle has a total of 360° In the circle shown, 35° are NOT shaded, so 325° ARE shaded

The fraction of the circle that is shaded is 332650 Multiply this fraction by the total area to find the shadedarea

17 Choice a is correct; log464 means 4?= 64; 43= 64 Therefore, log464 = 3

18 Choice j is correct The lines have the same y-intercept (b) Their slopes are opposites So, the slope of

the first line is m, thus, the slope of the second line is −m.

Since the y-intercept is b and the slope is −m, the equation of the line is y = −mx + b.

19 Choice b is correct Use the table below to organize the information.

Mark’s rate is 1 job in 40 minutes Audrey’s rate is 1 job in 50 minutes You don’t know how long it will

take them together, so time is x To find the work done, multiply the rate by the time.

Add the work done by Mark with the work done by Audrey to get 1 job done

The lessons in this chapter are intended to refresh your memory The 80 practice questions followingthese lessons contain examples of the topics covered here as well as other various topics you may see on theofficial ACT Assessment If in the course of solving the practice questions you find a topic that you are notfamiliar with or have simply forgotten, you may want to consult a textbook for additional instruction

• The math questions start easy and get harder Pace yourself accordingly.

• Study wisely The number of questions involving various algebra topics is significantly higher thanthe number of trigonometry questions Spend more time studying algebra concepts

• There is no penalty for wrong answers Make sure that you answer all of the questions, even ifsome answers are only a guess

• If you are not sure of an answer, take your best guess Try to eliminate a couple of the answerchoices

• If you skip a question, leave that question blank on the answer sheet and return to it when youare done Often, a question later in the test will spark your memory about the answer to a ques-tion that you skipped

• Read carefully! Make sure you understand what the question is asking

• Use your calculator wisely Many questions are answered more quickly and easily without a culator

cal-• Most calculators are allowed on the test However, there are some exceptions Check the ACTwebsite (ACT.org) for specific models that are not allowed

• Keep your work organized Number your work on your scratch paper so that you can refer back

to it while checking your answers

• Look for easy solutions to difficult problems For example, the answer to a problem that can besolved using a complicated algebraic procedure may also be found by “plugging” the answerchoices into the problem

• Know basic formulas such as the formulas for area of triangles, rectangles, and circles ThePythagorean theorem and basic trigonometric functions and identities are also useful, and not thatcomplicated to remember

The questions appear in order of difficulty on the test, but topics are mixed together throughout the test

Pre-Algebra

Topics in this section include many concepts you may have learned in middle or elementary school, such asoperations on whole numbers, fractions, decimals, and integers; positive powers and square roots; absolute

value; factors and multiples; ratio, proportion, and percent; linear equations; simple probability; using charts,tables, and graphs; and mean, median, mode, and range.

N UMBERS

■ Whole numbers Whole numbers are also known as counting numbers: 0, 1, 2, 3, 4, 5, 6,

■ Integers Integers are both positive and negative whole numbers including zero: −3, −2, −1, 0, 1,

13+ 27 The least common denominator for 3 and 7 is 21.

(13×77) + (27×33) Multiply the numerator and denominator of each fraction by the same

number so that the denominator of each fraction is 21

221+ 261= 281 Add the numerators and keep the denominators the same Simplify the

E XPONENTS AND S QUARE R OOTS

An exponent tells you how many times to the base is used as factor Any base to the power of zero is one

Make sure you know how to work with exponents on the calculator that you bring to the test Most

sci-entific calculators have a y x or x ybutton that is used to quickly calculate powers

When finding a square root, you are looking for the number that when multiplied by itself gives youthe number under the square root symbol

25 = 5

64 = 8

169 = 13

Have the perfect squares of numbers from 1 to 13 memorized since they frequently come up in all types

of math problems The perfect squares (in order) are:

F ACTORS AND M ULTIPLES

Factors are numbers that divide evenly into another number For example, 3 is a factor of 12 because it dividesevenly into 12 four times

6 is a factor of 66

9 is a factor of 27

−2 is a factor of 98

Multiples are numbers that result from multiplying a given number by another number For example,

12 is a multiple of 3 because 12 is the result when 3 is multiplied by 4

66 is a multiple of 6

27 is a multiple of 9

98 is a multiple of −2

R ATIO , P ROPORTION , AND P ERCENT

Ratios are used to compare two numbers and can be written three ways The ratio 7 to 8 can be written 7:8,

Percents are always “out of 100.” 45% means 45 out of 100 It is important to be able to write percents

as decimals This is done by moving the decimal point two places to the left

The probability of an event is P(event) =

For example, the probability of rolling a 5 when rolling a 6-sided die is 16, because there is one able outcome (rolling a 5) and there are 6 possible outcomes (rolling a 1, 2, 3, 4, 5, or 6) If an event is impos-sible, it cannot happen, the probability is 0 If an event definitely will happen, the probability is 1

favor-C OUNTING P RINCIPLE AND T REE D IAGRAMS

The sample space is a list of all possible outcomes A tree diagram is a convenient way of showing the sample

space Below is a tree diagram representing the sample space when a coin is tossed and a die is rolled

The first column shows that there are two possible outcomes when a coin is tossed, either heads or tails.The second column shows that once the coin is tossed, there are six possible outcomes when the die is rolled,numbers 1 through 6 The outcomes listed indicate that the possible outcomes are: getting a heads, thenrolling a 1; getting a heads, then rolling a 2; getting a heads, then rolling a 3; etc This method allows you toclearly see all possible outcomes

Another method to find the number of possible outcomes is to use the counting principle An example

of this method is on the following page

Coin

H

123456

Die Outcomes

H1H2H3H4H5H6

T

123456

T1T2T3T4T5T6

favorable



Nancy has 4 pairs of shoes, 5 pairs of pants, and 6 shirts How many different outfits can shemake with these clothes?

To find the number of possible outfits, multiply the number of choices for each item

4 × 5 × 6 = 120

She can make 120 different outfits

Helpful Hints about Probability

■ If an event is certain to occur, the probability is 1

■ If an event is certain NOT to occur, the probability is 0

■ If you know the probability of all other events occurring, you can find the probability of the remainingevent by adding the known probabilities together and subtracting that sum from 1

M EAN , M EDIAN , M ODE , AND R ANGE

Mean is the average To find the mean, add up all the numbers and divide by the number of items

Median is the middle To find the median, place all the numbers in order from least to greatest Count

to find the middle number in this list Note that when there is an even number of numbers, there will be twomiddle numbers To find the median, find the average of these two numbers

Mode is the most frequent or the number that shows up the most If there is no number that appearsmore than once, there is no mode

The range is the difference between the highest and lowest number

Example

Using the data 4, 6, 7, 7, 8, 9, 13, find the mean, median, mode, and range

Mean: The sum of the numbers is 54 Since there are seven numbers, divide by 7 to find themean 54 ÷ 7 = 7.71

Median: The data is already in order from least to greatest, so simply find the middle

num-ber 7 is the middle numnum-ber

Mode: 7 appears the most often and is the mode

Range: 13 − 4 = 9

L INEAR E QUATIONS

An equation is solved by finding a number that is equal to an unknown variable

Simple Rules for Working with Equations

1 The equal sign separates an equation into two sides.

2 Whenever an operation is performed on one side, the same operation must be performed on the other

side

3 Your first goal is to get all of the variables on one side and all of the numbers on the other.

4 The final step often will be to divide each side by the coefficient, leaving the variable equal to a

number

C ROSS -M ULTIPLYING

You can solve an equation that sets one fraction equal to another by cross-multiplication

Cross-multiplication involves setting the products of opposite pairs of terms equal

Special Tips for Checking Equations

1 If time permits, be sure to check all equations.

2 Be careful to answer the question that is being asked Sometimes, this involves solving for a variable

and than performing an operation

Example: If the question asks for the value of x − 2, and you find x = 2, the answer is not 2, but

2 − 2 Thus, the answer is 0

C HARTS , T ABLES , AND G RAPHS

The ACT Math Test will assess your ability to analyze graphs and tables It is important to read each graph

or table very carefully before reading the question This will help you to process the information that is sented It is extremely important to read all of the information presented, paying special attention to head-ings and units of measure Here is an overview of the types of graphs you will encounter:

pre-■ CIRCLE GRAPHS or PIE CHARTS

This type of graph is representative of a whole and is usually divided into percentages Each section of thechart represents a portion of the whole, and all of these sections added together will equal 100% of thewhole

■ BAR GRAPHS

Bar graphs compare similar things with bars of different length, representing different values These graphsmay contain differently shaded bars used to represent different elements Therefore, it is important to payattention to both the size and shading of the graph

Fruit Ordered by Grocer

100 80 60 40 20 0

Attendance at a Baseball Game

15%

girls24%

boys61%

adults

■ BROKEN LINE GRAPHS

Broken-line graphs illustrate a measurable change over time If a line is slanted up, it represents anincrease, whereas a line sloping down represents a decrease A flat line indicates no change

In the line graph below, Lisa’s progress riding her bike is graphed From 0 to 2 hours, Lisa movessteadily Between 2 and 212hours, Lisa stops (flat line) After her break, she continues again but at a slowerpace (line is not as steep as from 0 to 2 hours)

Elementary Algebra

Elementary algebra covers many topics typically covered in an Algebra I course Topics include operations onpolynomials; solving quadratic equations by factoring; linear inequalities; properties of exponents and squareroots; using variables to express relationships; and substitution

O PERATIONS ON P OLYNOMIALS

Combining Like Terms: terms with the same variable and exponent can be combined by adding the coefficients

and keeping the variable portion the same

S OLVING Q UADRATIC E QUATIONS BY F ACTORING

Before factoring a quadratic equation to solve for the variable, you must set the equation equal to zero

x2− 7x = 30

x2− 7x − 30 = 0

Lisa’s Progress

50 40 30 20 10 0

Time in Hours

1 2 3 4

Notice that the inequality switched from less than to greater than after division by a negative.

When graphing inequalities on a number line, recall that < and > use open dots and ≤ and ≥ use soliddots

Any number (or variable) to the zero power is 1.

U SING V ARIABLES TO E XPRESS R ELATIONSHIPS

The most important skill needed for word problems is being able to use variables to express relationships.The following will assist you in this by giving you some common examples of English phrases and their math-ematical equivalents

■ “Increase” means add

Example

A number increased by five = x + 5.

■ “Less than” means subtract

Example

10 less than a number = x − 10.

■ “Times” or “product” means multiply

Example

Three times a number = 3x.

■ “Times the sum” means to multiply a number by a quantity

Example

Five times the sum of a number and three = 5(x + 3).

■ Two variables are sometimes used together

When 14 is added to a number x, the sum is less than 21.

A SSIGNING V ARIABLES IN W ORD P ROBLEMS

It may be necessary to create and assign variables in a word problem To do this, first identify an unknownand a known You may not actually know the exact value of the “known,” but you will know at least some-thing about its value

Examples

Max is three years older than Ricky

Unknown = Ricky’s age = x

Known = Max’s age is three years older

Therefore,

Ricky’s age = x and Max’s age = x + 3

Siobhan made twice as many cookies as Rebecca

Unknown = number of cookies Rebecca made = x

Known = number of cookies Siobhan made = 2x

Cordelia has five more than three times the number of books that Becky has

Unknown = the number of books Becky has = x

Known = the number of books Cordelia has = 3x + 5

S UBSTITUTION

When asked to substitute a value for a variable, replace the variable with the value

Example

Find the value of x2+ 4x − 1, for x = 3.

Replace each x in the expression with the number 3 Then, simplify.

= (3)2+ 4(3) − 1

= 9 + 12 − 1

= 20

The answer is 20

Intermediate Algebra

Intermediate algebra covers many topics typically covered in an Algebra II course such as the quadratic mula; inequalities; absolute value equations; systems of equations; matrices; functions; quadratic inequali-ties; radical and rational expressions; complex numbers; and sequences

for-T HE Q UADRATIC F ORMULA

x = −b ±2b c a2− 4a for quadratic equations in the form ax2+ bx + c = 0.

The quadratic formula can be used to solve any quadratic equation It is most useful for equations that not be solved by factoring

can-A BSOLUTE V ALUE E QUATIONS

Recall that both |5| = 5 and |−5| = 5 This concept must be used when solving equations where the variable

is in the absolute value symbol

coor-Solve the following system of equations:

Next, multiply one of the equations so that the coefficient of one variable (we will use y) is

the opposite of the coefficient of the same variable in the other equation

−1(−x + y = 2)

2x + y = 17

x − y = −2

2x + y = 17

Add the equations One of the variables should cancel out.

Any number in the form a + bi is a complex number i = −1 Operations with i are the same as with any

variable, but you must remember the following rules involving exponents

5x−1x0= 5x((22))−1x0

= 120x−1x0

= 1x0

R ADICAL E XPRESSIONS

■ Radicals with the same radicand (number under the radical symbol) can be combined the same way

“like terms” are combined

In the fractional exponent, the numerator (top) is the power and the denominator (bottom) is the root

By representing radical expressions using exponents, you are able to use the rules of exponents to plify the expression

sim-I NEQUALITIES

The basic solution of linear inequalities was covered in the Elementary Algebra section Following are somemore advanced types of inequalities

Solving Combined (or Compound) Inequalities

To solve an inequality that has the form c < ax + b < d, isolate the letter by performing the same operation

on each member of the equation

Divide each term by −5, changing the direction of both inequality symbols:

−−55< −−55y < −205= 1 > y > −4

The solution consists of all real numbers less than 1 and greater than −4

Absolute Value Inequalities

|x | < a is equivalent to −a < x < a and |x| > a is equivalent to x > a or x < −a

Recall that quadratic equations are equations of the form ax2+ bx + c = 0.

To solve a quadratic inequality, first treat it like a quadratic equation and solve by setting the equationequal to zero and factoring Next, plot these two points on a number line This divides the number line intothree regions Choose a test number in each of the three regions and determine the sign of the equation when

it is the value of x Determine which of the three regions makes the inequality true This region is the answer.

Plot the numbers on a number line This divides the number line into three regions

The number line is divided into the following regions

numbers less than −3

numbers between −3 and 2

numbers greater than 2

−4 −3 −2 −1 0

numbers less than

−3

numbers between

−3 and 2

numbers greater than 2

1 2 3 4

Use a test number in each region to see if (x + 3)(x − 2) is positive or negative in that region numbers less than −3 numbers between −3 and 2 numbers greater than 2

(−5 + 3)(−5 − 2) = 14 (0 + 3 )(0 − 2) = −6 (3 + 3)(3 − 2) = 6

The original inequality was (x + 3)(x − 2) < 0 If a number is less than zero, it is negative The only region that is negative is between −3 and 2; −3 < x < 2 is the solution.

This section contains problems dealing with the (x, y) coordinate plane and number lines Included are slope,

distance, midpoint, and conics

1 1



The equation of a line is often written in slope-intercept form which is y = mx + b, where m is the slope and b is the y-intercept.

Important Information about Slope

■ A line that rises to the right has a positive slope and a line that falls to the right has a negative slope

■ A horizontal line has a slope of 0 and a vertical line does not have a slope at all—it is undefined

■ Parallel lines have equal slopes

= 1 where (h, k) is the center If the larger denominator

is under y, the y-axis is the major axis If the larger denominator is under the x-axis, the x-axis is the

major axis

Parabola y − k = a(x − h)2or x − h = a(y − k)2 The vertex is (h, k) Parabolas of the first form open

up or down Parabolas of the second form open left

trape-To begin this section, it is helpful to become familiar with the vocabulary used in geometry The listbelow defines some of the main geometrical terms:

you can draw in a circle

Hypotenuse the longest leg of a right triangle, always opposite the right angle

Volume of a rectangular solid V = lwh

B ASIC G EOMETRIC F ACTS

The sum of the angles in a triangle is 180°

A circle has a total of 360°

P YTHAGOREAN T HEOREM

The Pythagorean theorem is an important tool for working with right triangles.

It states: a2+ b2= c2, where a and b represent the legs and c represents the hypotenuse.

This theorem allows you to find the length of any side as along as you know the measure of the othertwo So, if leg a = 1 and leg b = 2 in the triangle below, you can find the measure of leg c

M ULTIPLES OF P YTHAGOREAN T RIPLES

Any multiple of a Pythagorean triple is also a Pythagorean triple Therefore, if given 3:4:5, then 9:12:15 is also

A right triangle with two angles each measuring 45 degrees is called an isosceles right triangle In an

isosce-les right triangle:

■ The length of the hypotenuse is 2 multiplied by the length of one of the legs of the triangle

■ The length of each leg is 22 multiplied by the length of the hypotenuse

45 °

45 °

x = y = 22 ×110= 1022 = 52

In a right triangle with the other angles measuring 30 and 60 degrees:

■ The leg opposite the 30-degree angle is half of the length of the hypotenuse (And, therefore, thehypotenuse is two times the length of the leg opposite the 30-degree angle.)

■ The leg opposite the 60-degree angle is 3 times the length of the other leg

Example

x = 2 · 7 = 14 and y = 73

C ONGRUENT

Two figures are congruent if they have the same size and shape

T RANSLATIONS , ROTATIONS , AND REFLECTIONS

Congruent figures can be made to coincide (place one right on top of the other), by using one of the followingbasic movements

Basic trigonometric ratios, graphs, identities, and equations are covered in this section

B ASIC T RIGONOMETRIC R ATIOS

sin A = hy o p p o p t o es n it u e se opposite refers to the length of the leg opposite angle A.

cos A = hy a p d o ja tec n en u t se adjacent refers to the length of the leg adjacent to angle A.

tan A = a o d p j p a oc s e i n te t

T RIGONOMETRIC I DENTITIES

sin2x + cos2x = 1

tan x = csionsx x

sin 2x = 2sin x cos x

cos 2x = cos2x − sin2x

tan 2x = 12−tatannx2x

 P r a c t i c e Q u e s t i o n s

Directions

After reading each question, solve each problem, and then choose the best answer from the choices given

(Remember, the ACT Math Test is different from all the other tests in that each math question contains five

answer choices.) When you are taking the official ACT, make sure you carefully fill in the appropriate ble on the answer document

bub-You may use a calculator for any problem, but many problems are done more quickly and easily out one

with-Unless directions tell you otherwise, assume the following:

■ figures may not be drawn to scale

■ geometric figures lie in a plane

■ “line” refers to a straight line

■ “average” refers to the arithmetic mean

Remember, the questions get harder as the test goes on You may want to consider this fact as you paceyourself

1 How is five hundred twelve and sixteen thousandths written in decimal form?

4 The ratio of boys to girls in a kindergarten class is 4 to 5 If there are 18 students in the class, how

many are boys?

6 Which of the following is NOT the graph of a function?